Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications
Guilherme Mazanti, Jaqueline G. Mesquita
TL;DR
This paper addresses the well-posedness and long-time behavior of the continuous-time delay difference equation $x(t)=A x(t-\tau(t))$ in $\mathbb{R}^d$ across continuous, regulated, and $L^p$ solution spaces. It introduces the notion of well-regulated delays, proves a representation formula $x(t)=A^{\mathbf n(t)} x_0(\sigma_{\mathbf n(t)}(t))$, and derives existence, uniqueness, and continuous dependence results, along with explicit exponential stability criteria. The analysis unifies spaces with different regularities, providing space-specific stability results and highlighting the robustness limitations with respect to delay variations. Applications to equations with state-dependent delays and to a transport equation via characteristics illustrate how the framework yields concrete exponential decay results in multiple norms. Overall, the work advances a rigorous, space-aware theory for delayed linear systems and related PDEs, with practical implications for control, dynamics with memory, and transport phenomena.
Abstract
In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - τ(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$, $A$ is a $d \times d$ matrix with real coefficients, and $τ\colon [0, +\infty) \to (0, +\infty)$ is a time-dependent delay. We provide our investigations for three spaces of functions: continuous, regulated, and $L^p$. We compare our results for these three cases, showing how the hypotheses change according to the space that we are treating. Finally, we provide applications of our results to difference equations with state-dependent delays for the cases of continuous and regulated function spaces, as well as to transport equations in one space dimension with time-dependent velocity.